Continuous Bundles of C*-Algebras with Discontinuous Tensor Products

For each non-exact C*-algebra A and infinite compact Hausdorffspace X there exists a continuous bundle B of C*-algebras onX such that the minimal tensor product bundle AB is discontinuous.The bundle B can be chosen to be unital with constant simplefibre. When X is metrizable, B can also be chosen to be separable.As a corollary, a C*-algebra A is exact if and only if A Bis continuous for all unital continuous C*-bundles B on a giveninfinite compact Hausdorff base space. The key to proving theseresults is showing that for a non-exact C*-algebra A there existsa separable unital continuous C*-bundle B on [0,1] such thatA B is continuous on [0,1] and discontinuous at 1, a counter-intuitiveresult. For a non-exact C*-algebra A and separable C*-bundleB on [0,1], the set of points of discontinuity of A B in [0,1]can be of positive Lebesgue measure, and even of measure 1.2000 Mathematics Subject Classification 46L06 (primary), 46L35(secondary).     

Decomposition Rank of Subhomogeneous C*-Algebras

We analyze the decomposition rank (a notion of covering dimensionfor nuclear C*-algebras introduced by E. Kirchberg and the author)of subhomogeneous C*-algebras. In particular, we show that asubhomogeneous C*-algebra has decomposition rank n if and onlyif it is recursive subhomogeneous of topological dimension n,and that n is determined by the primitive ideal space. As an application, we use recent results of Q. Lin and N. C.Phillips to show the following. Let A be the crossed productC*-algebra coming from a compact smooth manifold and a minimaldiffeomorphism. Then the decomposition rank of A is dominatedby the covering dimension of the underlying manifold. 2000 MathematicsSubject Classification 46L85, 46L35.     

Fourier Invariant Partially Approximating Subalgebras of the Irrational Rotation C*-Algebra

For a dense G-set of parameters, the irrational rotation algebrais shown to contain infinitely many C*-subalgebras satisfyingthe following properties. Each subalgebra is isomorphic to adirect sum of two matrix algebras of the same (perfect square)dimension; the Fourier transform maps each summand onto theother; the corresponding unit projection is approximately central;the compressions of the canonical generators of the irrationalrotation algebra are approximately contained in the subalgebra.2000 Mathematics Subject Classification 46L80, 46L40, 46L35.     

Entire cyclic homology of stable continuous trace algebras

A central result in this paper is the computation of the entirecyclic homology of canonical smooth subalgebras of stable continuoustrace C*-algebras having smooth manifolds M as their spectrum.More precisely, the entire cyclic homology is shown to be canonicallyisomorphic to the continuous periodic cyclic homology for thesealgebras. By an earlier result of the authors, one concludesthat the entire cyclic homology of the algebra is canonicallyisomorphic to the twisted de Rham cohomology of M.     

From Endomorphisms to Automorphisms and Back: Dilations and Full Corners

When S is a discrete subsemigroup of a discrete group G suchthat G = S^–1S, it is possible to extend circle-valuedmultipliers from S to G, to dilate (projective) isometric representationsof S to (projective) unitary representations of G, and to dilate/extendactions of S by injective endomorphisms of a C*-algebra to actionsof G by automorphisms of a larger C*-algebra. These dilationsare unique provided they satisfy a minimality condition. The(twisted) semigroup crossed product corresponding to an actionof S is isomorphic to a full corner in the (twisted) crossedproduct by the dilated action of G. This shows that crossedproducts by semigroup actions are Morita equivalent to crossedproducts by group actions, making powerful tools available tostudy their ideal structure and representation theory. The dilationof the system giving the Bost–Connes Hecke C*-algebrafrom number theory is constructed explicitly as an application:it is the crossed product C₀(A_f)Q*₊, corresponding to the multiplicativeaction of the positive rationals on the additive group A_f offinite adeles.     

Analytic Rank of Graph Algebras and Quantum Spaces

We show that the analytic rank, as defined by Murphy, of a unitalgraph C*-algebra is either 1 or 0, depending on whether or notthe underlying graph possesses an initial loop. As a corollary,we show that the analytic rank of certain quantum spaces (includingsome quantum spheres, projective spaces and lens spaces) is1. 2000 Mathematics Subject Classification 46L05, 46L65, 46L87.     

C*-Algebras Characterized by Ergodic Actions of the n-Torus

Using basic harmonic analysis on the n-torus, it is shown thatthe non-commutative n-tori are the only primitive C*-algebras(up to isomorphism) on which the n-torus Tⁿ acts freely andergodically. An explicit construction recovers the generatingunitaries in a natural way from the action. 2000 MathematicsSubject Classification 46L55.     

Reduced C*-algebras of Fell bundles over inverse semigroups

We construct a weak conditional expectation from the section C*-algebra of a Fell bundle over a unital inverse semigroup to its unit fibre. We use this to define the reduced C*-algebra of the Fell bundle. We study when the reduced C*-algebra for an inverse semigroup action on a groupoid by partial equivalences coincides with the reduced groupoid C*-algebra of the transformation groupoid, giving both positive results and counterexamples.     

Structural Projections on JBW*-Triples

A linear projection R on a Jordan*-triple A is said to be structuralprovided that, for all elements a, b and c in A, the equality{Rab Rc} = R{a Rbc} holds. A subtriple B of A is said to becomplemented if A = B + Ker(B), where Ker(B) = {aA: {B a B}= 0}. It is shown that a subtriple of a JBW*-triple is complementedif and only if it is the range of a structural projection. A weak* closed subspace B of the dual E* of a Banach space Eis said to be an N*-ideal if every weak* continuous linear functionalon B has a norm preserving extension to a weak* continuous linearfunctional on E* and the set of elements in E which attain theirnorm on the unit ball in B is a subspace of E. It is shown thata subtriple of a JBW*-triple A is complemented if and only ifit is an N*-ideal, from which it follows that complemented subtriplesof A are weak* closed, and structural projections on A are weak*continuous and norm non-increasing. It is also shown that everyN*-ideal in A possesses a triple product with respect to whichit is a JBW*-triple which is isomorphic to a complemented subtripleof A.     

Covariant Representations for Coactions of Hopf C*-Algebras

Given a coaction of a Hopf C*-algebra A on a C*-algebra B withan -invariant C*-subalgebra C, and a conditional expectationE: B C commuting with , it is shown that if (, u) is a covariantrepresentation of the system (C, A, |_C), then there is an associatedcovariant representation of the system (B, A, ), where is the representation induced from up to B via E, and is a unitary corepresentation of Anaturally associated with u. Some applications are also discussed,including a lifting of ergodic coactions to von Neumann algebras,and a characterization of the amenability of multiplicativeunitary operators via infinite tensor product covariant representations.2000 Mathematics Subject Classification 46L55, 46L60, 46L87,46L89, 81R15, 81R50, 81T05.     

C*-Algebras with Dense Nilpotent Subalgebras

In a beautiful result, Herrero (D. A. Herrero, ‘Normallimits of nilpotent operators’, Indiana Univ. Math. J.23 (1973/74) 1097–1108) showed that a normal operatoron l² lies in the closure of the set of nilpotent operatorsif and only if its spectrum is connected and contains zero.In the quest for an automatic continuity result for algebrahomomorphisms between C* -algebras, Dales showed that, if adiscontinuous algebra homomorphism : A u exists between C*-algebrasA and u, and if (A) is dense in u, then there is a C*-algebrau₂ with a dense subalgebra N u² such that every x N is quasinilpotent(see p. 685 of H. G. Dales, Banach algebras and automatic continuity,London Mathematical Society Monographs 24, Oxford UniversityPress, 2001). (A discontinuous homomorphism ₂: A₂ u₂ can bedefined with the same basic properties as , but the revisedtarget space u₂ has a dense subalgebra consisting of quasinilpotentelements.) As remarked by Dales, no such C*-algebra was thenknown; but here we present one. Indeed, using the full powerof Herrero's result, one may arrange that every x N is nilpotent.The C*-algebra is constructed in a ‘neat’ way; itis most naturally constructed as a non-separable, concrete C*-algebraof operators on a separable Hilbert space K but one can arrangethat the algebra u itself be separable if desired. 2000 MathematicsSubject Classification 47C15, 46H40 (primary), 47A10, 46L06,46L05, 46H35 (secondary).     

The Tracial Topological Rank of C*-Algebras

We introduce the notion of tracial topological rank for C*-algebras.In the commutative case, this notion coincides with the coveringdimension. Inductive limits of C*-algebrasof the form PM_n(C(X))P,where X is a compact metric space with dim X k, and P is aprojection in M_n(C(X)), have tracial topological rank no morethan k. Non-nuclear C*-algebras can have small tracial topologicalrank. It is shown that if A is a simple unital C*-algebra withtracial topological rank k (< ), then

(i) A is quasidiagonal,

(ii) A has stable rank 1,

(iii) A has weakly unperforatedK₀(A),

(iv) A has the following Fundamental Comparabilityof Blackadar:if p, q A are two projections with (p) < (q)for all tracialstates on A, then p q

. 2000 MathematicsSubject Classification: 46L05, 46L35.     

Jensen's Operator Inequality

Jensen's operator inequality and Jensen's trace inequality forreal functions defined on an interval are established in whatmight be called their definitive versions. This is accomplishedby the introduction of genuine non-commutative convex combinationsof operators, as opposed to the contractions considered in earlierversions of the theory by the authors, and by Brown and Kosaki.As a consequence, one no longer needs to impose conditions onthe interval of definition. It is shown how this relates tothe pinching inequality of Davis, and how Jensen's trace inequalitygeneralizes to C*-algebras. 2000 Mathematics Subject Classification47A56 (primary), 46L10, 47C15, 26A51 (secondary).     

Dual Pairs of Hopf *-Algebras

If A is a Hopf *-algebra, the dual space A' is again a *-algebra.There is a natural subalgebra A° of A' that is again a Hopf*-algebra. In many interesting examples, A° will be largeenough (to separate points of A). More generally, one can considera pair (A, B) of Hopf *-algebras and a bilinear form on A xB with conditions such that, if the pairing is non-degenerate,one algebra can be considered as a subalgebra of the dual ofthe other. In these notes, we study such pairs of Hopf *-algebras. We startfrom the notion of a Hopf *-algebra A and its reduced dual A°.We give examples of pairs of Hopf *-algebras, and discuss theproblem of non-degeneracy. The first example is an algebra pairedwith itself. The second example is the pairing of a Hopf *-algebra(due to Jimbo) and the twisted SU(n) of Woronowicz. We alsodiscuss the notion of the quantum double of Drinfeld in thisframework of dual pairs.     

Commutative Nilpotent Extensions of Commutative C*-Algebras Split

It is proved that every commutative, nilpotent extension ofa commutative C*-algebra splits topologically. 1991 MathematicsSubject Classification 46J40.     

On Holomorphic Principal Bundles Over a Compact Riemann Surface Admitting a Flat Connection, II

Holomorphic principal bundles over a compact Riemann surfaceX that admits a flat connection are considered. A holomorphicG-bundle over X, where G is a connected semisimple linear algebraicgroup over C, admits a flat connection if and only if the adjointvector bundle admits one. More generally, for a complex reductivegroup G, the necessary and sufficient condition on a G-bundleto admit a flat connection is described. This simplifies thecriterion obtained by the authors and given in Math. Ann. 322(2002) 333–346. 2000 Mathematics Subject Classification53C05, 32L05.     

The Operator Hilbert Space OH and Type III Von Neumann Algebras

A proof is given to show that the operator Hilbert space OHdoes not embed completely isomorphically into the predual ofa semi-finite von Neumann algebra. This complements Junge'srecent result, which admits such an embedding in the non-semi-finitecase. 2000 Mathematics Subject Classification 46L07, 46L54,47L25, 47L50.     

Absolutely continuous representations and a Kaplansky density theorem for free semigroup algebras

We introduce notions of absolutely continuous functionals and representations on the non-commutative disk algebra A_n. Absolutely continuous functionals are used to help identify the type L part of the free semigroup algebra associated to a *-extendible representation σ. A *-extendible representation of A_n is regular if the absolutely continuous part coincides with the type L part. All known examples are regular. Absolutely continuous functionals are intimately related to maps which intertwine a given *-extendible representation with the left regular representation. A simple application of these ideas extends reflexivity and hyper-reflexivity results. Moreover the use of absolute continuity is a crucial device for establishing a density theorem which states that the unit ball of σ(A_n) is weak-* dense in the unit ball of the associated free semigroup algebra if and only if σ is regular. We provide some explicit constructions related to the density theorem for specific representations. A notion of singular functionals is also defined, and every functional decomposes in a canonical way into the sum of its absolutely continuous and singular parts.     

Strong Operator Modules and the Haagerup Tensor Product

Strong operator modules over a von Neumann algebra R are introducedand the so-called extended Haagerup tensor product over R ofstrong modules is studied. In the case R=C and the spaces areweak* closed this product agrees with the weak* Haagerup tensorproduct of Blecher and Smith. If C is the center of R, the extendedHaagerup tensor product R is a Banach algebra containing thecentral tensor product of Chatterjee and Smith and has very nice properties concerningslice maps and the relative commutants of subalgebras. It isshown that R is a dual Banach space, and all weak* closed two-sidedideals of the algebra R are determined. 1991 Mathematics SubjectClassification: 46L05